THE SHAPE AND THE SIZE OF THE EARTH.
First Approximation.— The form and the dimensions of the earth have presented a prob lem of the greatest interest and difficulty to men of science for more than 20 centuries and they promise to tax the resources of the ablest physicists for some centuries to come. The first approximation to a solution of this prob lem, of which we have definite record, was made by Eratosthenes of the Alexandrian school of astronomers. Assuming the surface of the earth to be spherical, he measured the length and the angular amplitude of an arc of a great circle extending from Alexandria to Syene. He observed that at Syene, which is about 500 miles south of Alexandria, the sun shone vertically downwards into deep wells at noon on the day of the summer solstice, prov ing thus that at that place and time the sun was in the zenith. On the same day at Alex andria he observed, by means of the gnomon, that the sun at noon was south of the zenith by one-fiftieth of a circle, or 72 degrees. The principles involved in these measurements and in the calculation of the size of the earth are the earth are sensibly parallel. Thus, in Fig. 1, if A and B indicate the relative positions of Alexandria and Syene and C the centre of the earth, lines from A and C to the sun will be parallel; and hence the angle ZAS, or the meridian zenith distance of the sun at Alexan dria, will be equal to the angle ACB. Know ing this angle and the distance AB, the rule of three gives the entire circumference.
Second substantial ad vance beyond this first approximation was made until Newton showed that the gravitation and the rotation of the earth ought to make it somewhat flattened at the poles, or that the surface of the earth should have the shape of an oblate spheroid of revolution. The proof of this theoretical conclusion of Newton con stitutes the second approximation to the figure and the size of the earth. Such a figure, how ever, is much more difficult of measurement than a spherical figure. This is seen by a glance
at Fig. 2 representing an ellipse, which, if re volved about its shorter axis PP', will gener ate an oblate spheroidal surface. The prin ciples of mechanics show that when such a sur face is due to the attraction and the rotation of a fluid mass, the plumb line at any place will not in general point toward the centre of the mass, but will pass somewhat to one side of it as shown by the line LQ in the figure.
Newton's conception, therefore, involved th difficulties of the more complex spheroidal fig ure and of the hypothesis that the earth wa primatively a fluid mass. Two ways of testini Newton's views were proposed_ One was t very simple, but they are so fundamental as to justify a full explanation. They assume, first, that the earth is spherical in shape; sec ondly, that the plumb-bob at any point of the earth's surface is directed toward the earth's centre; and, thirdly, that the sun is so distant that lines drawn to it from different parts of measure the meridional lengths of a degree of latitude at different places on the earth's sur face. If the earth is an oblate spheroid, it is seen from Fig. 3 that the meridional distance along the surface intercepted by two plumb lines which make an angle of one degree (or any constant angle) with each other is greater at the poles than at the equator, or in general, greater in high than in low latitudes. The other method proposed to measure by means of the pendulum the varying acceleration to which a body is subject in different latitudes on the earth's surface. If the Newtonian view is cor rect, that acceleration, which is the resultant of the effects of attraction and rotation of the earth, and hence the weight of a body, must in crease in passing from the equator to the poles.
those derived from computations carried out to the nearest foot. More approximate values will be determined from computations now in prog ress, and they may possibly show Clarke's val ues to be in error by a few hundred feet.